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valsi 
cnansari 
type 
fu'ivla 
creator 
krtisfranks 
time entered 
Sun Apr 1 05:01:25 2018 
Examples


English


Definition #70583
 Preferred
[edit]


definition 
x_{1} is the meanvalue theorem mean/forwarddifferencequotient mean of the elements of (multi)set x_{2} (1element or 2element set) under/for function x_{3}. 
notes 
Let the sumti of x_{3} be a realvalued, univariate function f which is defined and continuous on at least the closed interval [
min(x_{2}), max(x_{2})] and differentiable on at least the open interval
(min(x_{2}), max(x_{2})), such that the derivative f' is injective. Then
x_{1} = (f')^{(1)}((f (max(x_{2}))  f (min(x_{2})))/(max(x_{2})  min(x_{2}))) if x_{2} has cardinality 2; if x_{2} has cardinality 1, then x_{1} is equal to the sole element of that set; other repairs may be necessary, depending on f: for example, f = log has x_{1} = 0 if 0 is an element of x_{2}. If x_{3} is filled with the monic, singleterm polynomial of (integer) degree p such that p > 1, then x_{1} will be the principal (p1)th root of the (1/p)scaled weighted (p,1)Lehmer mean (cnanlime) of the elements of x_{2} wherein the weight of the lesser of them is 1 (and the other weight is +1) unless x_{2} is a singleton (in which case the weights are both +1). Usually, x_{1} is the value which is guaranteed to exist for appropriate function x_{2} by the MeanValue Theorem.

gloss words 

created by 
krtisfranks 
vote information 
1

time 
Sun Apr 1 05:59:29 2018 



Examples




